$\overline{AB} = 5\sqrt{2}$ $\overline{BC} = {?}$ $A$ $C$ $B$ $5\sqrt{2}$ $?$ $ \sin( \angle ABC ) = \frac{ \sqrt{2}}{2}, \cos( \angle ABC ) = \frac{ \sqrt{2}}{2}, \tan( \angle ABC ) = 1$
$\overline{AB}$ is the hypotenuse $\overline{BC}$ is adjacent to $\angle ABC$ SOH CAH TOA We know the hypotenuse and need to solve for the adjacent side so we can use the cos function (CAH) $ \cos( \angle ABC ) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{\overline{BC}}{\overline{AB}}= \frac{\overline{BC}}{5\sqrt{2}} $ $ \overline{BC}=5\sqrt{2} \cdot \cos( \angle ABC ) = 5\sqrt{2} \cdot \frac{ \sqrt{2}}{2} = 5$